An Introduction to Probability and Stochastic Processes

227 Pages·1993·11.05 MB·English

by  Marc A. Berger (auth.)

Springer Texts in Statistics
Advisors:
Stephen Fienberg Ingram Olkin
Springer Texts in Statistics
Alfred Elements of Statistics
for the Life and Social Sciences
Berger An Introduction to Probability
and Stochastic Processes
Blom Probability and Statistics:
Theory and Applications
Chow and Teicher Probability Theory: Independence,
Interchangeability, Martingales
Second Edition
Christensen Plane Answers to Complex Questions:
The Theory of Linear Models
Christensen Linear Models for Multivariate, Time
Series, and Spatial Data
Christensen Log-Linear Models
du Toit, Steyn and Graphical Exploratory Data Analysis
Stumpf
Finkelstein and Levin Statistics for Lawyers
Jobson Applied Multivariate Data Analysis,
Volume I: Regression and Experimental
Design
Jobson Applied Multivariate Data Analysis,
Volume II: Categorical and Multivariate
Methods
Kalbfleisch Probability and Statistical Inference:
Volume 1: Probability
Second Edition
Kalbfleisch Probability and Statistical Inference:
Volume 2: Statistical Inference
Second Edition
(continued after index)
Marc A. Berger
An Introduction to
Probability and
Stochastic Processes
With 19 Illustrations
Springer-Verlag
New Yark Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Marc A. Berger
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0269
USA
Editorial Board
Stephen Fienberg Ingram Olkin
York University Department of Statistics
North York, Ontario N3J IP3 Stanford University
Canada Stanford, CA 94305
USA
Cover illustration: A zoom-in view of a fractal dragon set. This image is generated from a 2-map
iterated function system. (See color plate 11.)
Mathematics Subject Classifications (1991): 60-01, 60E05, 60110, 60127, 60E1O, 60F05, 60F15,
60GI0,47A35
Library of Congress Cataloging-in-Publication Data
Berger, Marc A., 1955-
An introduction to probability and stochastic processes / Marc A.
Berger.
p. cm.
Includes bibliographical references and index.
ISBN·13:978-1-4612-7643-2
1. Probabilities. 2. Stochastic processes. I. Title.
CA273.B48 1992
519.2-dc20 91-43019
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© 1993 Springer-Verlag New York, Inc.
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ISBN·13:978-1-4612-7643-2 e-ISBN-I3:978-1-4612-2726-7
DOl: 10.1007/978-1-4612-2726-7
In memory of Dr. Robert A. Dannels, a great man and a
dear friend who always provided advice and encourage
ment at critical times-with his unique touch of humor.
Preface
These notes were written as a result of my having taught a "nonmeasure
theoretic" course in probability and stochastic processes a few times at
the Weizmann Institute in Israel. I have tried to follow two principles.
The first is to prove things "probabilistically" whenever possible without
recourse to other branches of mathematics and in a notation that is as
"probabilistic" as possible. Thus, for example, the asymptotics of pn for
large n, where P is a stochastic matrix, is developed in Section V by using
passage probabilities and hitting times rather than, say, pulling in Perron
Frobenius theory or spectral analysis. Similarly in Section II the joint
normal distribution is studied through conditional expectation rather
than quadratic forms.
The second principle I have tried to follow is to only prove results
in their simple forms and to try to eliminate any minor technical com
putations from proofs, so as to expose the most important steps. Steps
in proofs or derivations that involve algebra or basic calculus are not
shown; only steps involving, say, the use of independence or a dominated
convergence argument or an assumptjon in a theorem are displayed. For
example, in proving inversion formulas for characteristic functions I omit
steps involving evaluation of basic trigonometric integrals and display
details only where use is made of Fubini's Theorem or the Dominated
Convergence Theorem. Similarly in proving the Berry-Esseen bound I
leave out steps where algebraic substitution and simplification are carried
out. This was done to make the proofs more transparent and to not
clutter up the arguments.
In many texts, generalized versions ofthe limit theorems are presented,
and as a result the proofs get involved. My point in these notes is to show
that the basic limit results can be proved concisely. For example, in
viii Preface
proving the Strong LLN some authors will first show that a.s. conver
gence and convergence in probability are equivalent for sums of indepen
dent random variables. Confining myself to proving the basic Strong
Law, I only use Kolmogorov's Maximal Inequality, like in Billingsley
[4]; as a result the proof fits into two to three pages. Similarly many
authors prove the Berry-Esseen bound for Lindeberg-Feller versions of
the CLT . I confine myself to proving this bound for the basic CLT (in its
original form) as a result of which the proof, taken from Feller [17], is
rather quick.
The notes here are designed to fill a one-semester course. I have tried to
make them complete and self-contained. With just a few exceptions every
result needed is proved herein rather than quoted from outside. Among
the exceptions are the Dominated Convergence Theorem, Fubini's Theo
rem, Abel's Lemma, and the fact that the Legendre-Fenchel transform of
a strictly convex function is essentially differ.enJiable. My feeling in these
cases was that effort in these results would both sidetrack too much and
push the notes out of the range of a single semester. I also did not prove
the limit result for P(t) as t --+ 00, for Markov pure jump processes.
Because of the brevity of these notes it is important to supplement details
about background and perspective. Thus, for example, before studying
extremes and extremal distributions in Section III, where the Poisson
distribution comes in, one should first discuss order statistics, where the
binomial distribution comes in.
The material in these notes can be grouped into four categories. The
first two sections deal with the theory of random variables and distribu
tions. The third section covers the basic limit theorems. The fourth, fifth
and sixth sections cover discrete and continuous time Markov processes.
Finally, the last section covers products of random matrices and their
application to generation of fractals. As mentioned above, these notes are
intended to be an instructional guide, and as such the material contained
herein was merged together from many different sources. Some of the
presentation is my own, but most of it comes from the references listed. I
drew most heavily from Billingsley [4], Breiman [5] and Feller [17] for
the first three sections and from Hoel, Port and Stone [28], Karlin and
Taylor [31] and Parzen [45] for the next three sections. Most of the
collected exercises also come from these authors. (Incidentally they vary
greatly in levels of difficulty.)
In the most concise way, my advice to anyone teaching "nonmeasure
theoretic" probability at a serious level is to emulate Feller's marvelous
volumes as much as possible. Much of my excitement about the field of
probability theory stems from his writings.
Acknowledgments
for Permissions
Springer-Verlag wishes to thank the publishers listed below for their
copyright permission and endorsement to use their previously published
material in this book. Their invaluable help in this matter has made the
publication of this volume possible.
Exercises 10 in Chapter 2; 19, 25, 27, and 28 in Chapter 4; 8, 9,
10,19, and 20 in Chapter 5; and 4, 7, 8, 9,18,19, and 20 in Chapter 6 have
been reproduced with the kind permission of Academic Press Inc. from
A First Course in Stochastic Processes (Second Ed.) by S. Karlin and
H.M. Taylor.
Material in this book appeared in Sections 1.2 and 1.3 (pages 7 -12) and
Exercise 13 in Section 9.1 (page 306) in A Course in Probability Theory
by K.L. Chung, and has been reproduced with the kind permission of
Academic Press Inc. and K.L. Chung.
Exercises 2 and 3 in Chapter 2 has been reprinted with permission from
Probability, Classics in Applied Probability, Number 7, by Leo Breiman.
Copyright 1992 by the Society for Industrial and Applied Mathematics,
Philadelphia, Pennsylvania. All rights reserved.
Exercises 5, 6, 7, and 8 in Chapter 1; 11 in Chapter 2; and 2, 3
and 4 in Chapter 3 have been reproduced with the kind permission of
Chelsea Publishing Company from The Theory of Probability by B.V.
Gnedenko.
Exercises 1,2,3, and 4 in Chapter 1; 1,4,5,6,7,8,12,13,14,15,16,17,
and 18 in Chapter 2; l(a-i), 1( k), 7,18 and 26 in Chapter 4; l(a-i), l(k),
3, 6, and 7 in Chapter 5; and 11, 12, 13, and 14 in Chapter 6 have
been reproduced with the kind permission of E. Parzen from Stochastic
Processes by E. Parzen. Also, material in this book appeared in Chapters
6 and 7 of Stochastic Processes.
x Acknowledgments for Permissions
Exercises 1 (j), 1( 1), 2, 3, 4, 5, 6, 11, 12, 13, 14, 15, 16, 17, 21, and 23 in
Chapter 4; 1(j), 1(1),2,4,5,11, 13, 14, 17, and 18 in Chapter 5; and 1,2,
5, 6, 15, and 16 in Chapter 6 have been reproduced with the kind per
mission of Houghton Mifflin Publishing Company from Introduction
to Stochastic Processes by P.G. Hoel, S.c. Port and c.J. Stone. Also,
material in Chapters 4-6 of this book appeared in Chapters 1-3 of
Hoel, P.G., S.c. Port, and c.J. Stone, Introduction to Stochastic Pro
cesses. Copyright © 1972 by Houghton Mifflin Company. Used with
permission.
Material in this book appeared on pages 219-222 of Probability THeory
I (4th Edition) by M. Loeve, and has been reproduced with the kind
permission of Springer-Verlag.
Material in this book appeared in Sections 3 and 4 of Large Deviations
and Applications by S.R.S. Varadhan. Reprinted with permission from
the CBMS-NSF Regional Conference Series in Applied Mathematics,
Number 46. Copyright 1984 by the Society for Industrial and Applied
Mathematics, Philadelphia, Pennsylvania. All rights reserved.
Material in this book appeared in Example XV.2(j) on page 380, and
on pages 452-453 in An Introduction to Probability Theory and Its Appli
cations, Volume I (3rd ed.) by W. Feller, © 1968. Reprinted by permission
of John Wiley & Sons, Inc.
Material in this book appeared in Section 22 (pages 248-251) of
Probability and Measure by P. Billingsley, © 1979. Reprinted by per
mission of John Wiley & Sons, Inc.
Contents
Preface ............... . vii
Acknowledgments for Permissions ix
I. Univariate Random Variables 1
Discrete Random Variables 1
Properties of Expectation ...... . 2
Properties of Characteristic Functions ..... . 6
Basic Distributions ......... . 8
Absolutely Continuous Random Variables . 11
Basic Distributions ............. . 16
Distribution Functions ........... . 20
Computer Generation of Random Variables 23
Exercises .......... . ...... . 24
II. Multivariate Random Variables 27
Joint Random Variables ..... . 27
Conditional Expectation . . . . . . 33
Orthogonal Projections . . . . . . . 36
Joint Normal Distribution ........ . 38
Multi-Dimensional Distribution Functions 39
Exercises 42
III. limit Laws 45
Law of Large Numbers 45
Weak Convergence 48
Bochner's Theorem 58